Ch11hw1 - 614 Kinematics of Particles Fig P11.9 and P11.10...

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Unformatted text preview: 614 Kinematics of Particles Fig. P11.9 and P11.10 Fig. P11.17 and P11.18 11.7 The motion of a particle is deﬁned by the relation x = 2t3 — 12t2 - 72t — 80, where x and t are expressed in meters and seconds, re- spectively. Determine (a) when the velocity is zero, (19) the velocity, the ac- celeration, and the total distance traveled when x = 0. 11.8 The motion of a particle is deﬁned by the relation x = 2t3 — 18t2 + 48t — 16, where x and t are expressed in millimeters and seconds, re- spectively. Determine (a) when the velocity is zero, (1)) the position and the total distance traveled when the acceleration is zero. 11.9 The acceleration of point A is deﬁned by the relation (1 = —l.8 sin kt, where a and t are expressed in m/s2 and seconds, respectively, and k = 3 rad/s. Knowing that x = 0 and v = 0.6 m/s when t = 0, determine the velocity and position of point A when t = 0.5 s. 1 1 .10 The acceleration of point A is deﬁned by the relation (1 = — 1.08 sin kt — 1.44 cos kt, where a and t are expressed in m/s2 and seconds, re- spectively, and k = 3 rad/s. Knowing that x = 0.16 m and v = 0.36 m/s when t = 0, determine the velocity and position of point A when t = 0.5 5. 11.11 The acceleration of a particle is directly proportional to the square of the time t. When t = 0, the particle is at x = 36 ft. Knowing that at t = 9 s, x = 144 ft and v = 27 ft/s, express 9: and o in terms of t. 11.12 It is known that from t = 2 s to t = 10 s the acceleraﬁon of a particle is inversely proportional to the cube of the time t. When t = 2 s, v = —15 ft/s, and when t = 10 s, v = 0.36 ft/s. Knowing that the particle is twice as far from the origin when t = 2 s as it is when t = 10 s, determine (a) the position of the particle when t = 2 s and when t = 10 s, (b) the total distance traveled by the particle from t = 2 s to t = 10 s. 1 1. 13 The acceleration of a particle is directly proportional to the time t. At t = 0, the velocity of the particle is 400 mm/s. Knowing that v = 370 mm/s and x = 500 mm when t = l 5, determine the velocity, the posi— tion, and the total distance traveled when t = 7 s. 11.14 The acceleration of a particle is deﬁned by the relation a = 0.15 m/sz. Knowing that x = —10 m when t = 0 and v = —0.15 m/s when t = 2 5, determine the velocity, the position, and the total distance traveled when t = 5 5. 11.15 The acceleration of point A is deﬁned by the relation (1 = 200x(l + kxz), where a and x are expressed in m/s2 and meters, respectively, and k is a constant. Knowing that the velocity of A is 2.5 m/s when x = 0 and 5 m/s when x = 0.15 m, determine the value of k. 11.16 The acceleration of point A is deﬁned by the relation (1 = 200x + 3200x3, where a and x are expressed in rn/s2 and meters, respectively. Knowing that the velocity of A is 2.5 m/s and x = 0 when t = 0, determine the velocity and position of A when t = 0.05 5. 11.17 Point A oscillates with an acceleration a = 2880 — 144x, where a and x are expressed in in./s2 and inches, respectively. The magnitude of the velocity is 11 in./s when x = 20.4 in. Determine (a) the maximum velocity of A, (b) the two positions at which the velocity of A is zero. 11.18 Point A oscillates with an acceleration a = 144(20 — x), where a and x are expressed in in./s2 and inches, respectively. Knowing that the sys- tem starts at time t = 0 with v = 0 and x = 19 in., determine the position and the velocity of A when t = 0.2 s. re- ac- re- the - 1.8 and the 1.08 then time posi- 1 a = when veled 1 a = tively, O and 1 a = tively. rmine where of the city of where ne sys— osition 11.19 The acceleration of a particle is deﬁned by the relation a = 1296 — 28, where a and x are expressed in m/s2 and meters, respectively. Knowing that v = 8 m/s when x = 0, determine (a) the maximum value ofx, (b) the velocity when the particle has traveled a total distance of 3 m. 11.20 The acceleration of a particle is deﬁned by the relation (1 = k(l — e‘x), where k is a constant. Knowing that the velocity of the particle is o = +9 m/s when x = —3 m and that the particle comes to rest at the origin, determine (a) the value of k, (b) the velocity of the particle when x = —2 m. 11.21 The acceleration of a particle is deﬁned by the relation (1 = —k\/5, where k is a constant. Knowing that x = 0 and v = 25 ft/s at t = 0, and that v = 12 ft/s when x = 6 ft, determine (a) the velocity of the particle when x = 8 ft, (19) the time required for the particle to come to rest. 11.22 Starting from x = 0 with no initial velocity, a particle is given an acceleration a = 08sz + 49, where a and v are expressed in ft/s2 and ft/s, respectively. Determine (a) the position of the particle when u = 24 ft/s, (b) the speed of the particle when x = 40 ft. 11.23 The acceleration of slider A is deﬁned by the relation (1 = —2ka2 — 02, where a and v are expressed in m/s2 and m/s, respectively, and k is a constant. The system starts at time t = 0 with x = 0.5 m and v = 0. Knowing that x = 0.4 in when t = 0.2 5, determine the value of k. 11.24 The acceleration of slider A is deﬁned by the relation (1 = -2 V 1 — 02, where a and v are expressed in m/s2 and m/s, respectively. The system starts at time t = 0 with x = 0.5 m and v = 0. Determine (a) the position ofA when u = —0.8 m/s, (1)) the position ofA when t = 0.2 5. 11.25 The acceleration of a particle is deﬁned by the relation (1 = —kv2'5, where k is a constant. The particle starts at x = 0 with a velocity of 16 mm/s, and when x = 6 mm the velocity is observed to be 4 mm/s. De~ termine (a) the velocity of the particle when x = 5 mm, (b) the time at which the velocity of the particle is 9 mm/s. 11.26 The acceleration of a particle is deﬁned by the relation (1 = 0.6(1 — k0), where k is a constant. Knowing that at t = 0 the particle starts from rest at x = 6 m and that v = 6 m/s when t = 20 5, determine (a) the constant k, (b) the position of the particle when u = 7.5 m/s, (0) the maxi- mum velocity of the particle. 11.27 Experimental data indicate that in a regiOn downstream of a given louvered supply vent the velocity of the emitted air is deﬁned by v = 0.18vo/x, where v and x are expressed in ft/s and feet, respectively, and 00 is the initial discharge velocity of the air. For 120 = 12 ft/s, determine (a) the acceleration of the air at x = 6 ft, ([9) the time required for the air to ﬂow fromx=3fttox= 10ft. 11.28 Based on observations, the s eed of a jogger can be approx— imated by the relation 1) = 7.5(1 — 0.04x) '3, where v and x are expressed in km/h and kilometers, respectively. Knowing that x = 0 at t = 0, deter— mine (a) the distance the jogger has run when t = 1 h, (b) the jogger’s acceleration in m/52 at t = 0, (c) the time required for the jogger to run 6 km. Fig. P11.28 Problems 61 5 616 Kinematics of Particles Fig. P11.29 P ‘1" I: | I | l | | Fig. P11.30 11.29 The acceleration due to gravity of a particle falling toward the earth is a = —gR2/1‘2, where r is the distance from the center of the earth to the particle, H is the radius of the earth, and g is the acceleration due to grav- ity at the surface of the earth. If R = 3960 mi, calculate the escape velocity, that is, the minimum velocity with which a particle must be projected upward from the surface of the earth if it is not to return to earth. (Hint: o = 0 for r = <30.) 11.30 The acceleration due to gravity at an altitude y above the sur- face of the earth can be expressed as . A. [1 + (y/20.9 x 106)]2 d— where a and y are expressed in ft/s2 and feet, respectively. Using this expression, compute the height reached by a projectile ﬁred vertically upward from the surface of the earth if its initial velocity is (a) o = 2400 ft/s, (b) o = 4000 ft/S, (c) 1) = 40,000 ft/S. 1 1 .31 The velocity of a slider is deﬁned by the relation 0 = o’ sin(wnt + 4)). Denoting the velocity and the position of the slider at t = 0 by 00 and x0, respectively, and knowing that the maximum displacement of the slider is 2x0, show that (a) o’ = (00 + xgwﬁVZxown, (b) the maximum value of the ve- locity occurs when x = x0[3 - (co/xown)2] / 2. 1 1 .32 The velocity of a particle is o = oo[1 — sin(1Tt/T)]. Knowing that the particle starts from the origin with an initial velocity 00, determine (a) its position and its acceleration at t = 3T, (22) its average velocity during the intervalt=0tot =T. ________—_————— 11.4. UNIFORM RECTILINEAR MOTION Uniform rectilinear motion is a type of straight—line motion which is frequently encountered in practical applications. In this motion, the acceleration a of the particle is zero for every value of t. The veloc- ity o is therefore constant, and Eq. (11.1) becomes dx — = v = constant dt The position coordinate x is obtained by integrating this equation. De— noting by x0 the initial value of x, we write fdx=vfotdt x—x0=vt x = x0 + vt (ll-5) This equation can be used only if the velocity of the particle is known to be constant. 11.37 A group of students launches a model rocket in the vertical di- rection. Based on tracking data, they determine that the altitude of the rocket was 27.5 m at the end of the powered portion of the ﬂight and that the rocket landed 16 5 later. Knowing that the descent parachute failed to deploy so that the rocket fell freely to the ground after reaching its maximum altitude and assuming that g = 9.81 m/sz, determine (a) the speed 01 of the rocket at the end of powered ﬂight, (19) the maximum altitude reached by the rocket. 11.38 A sprinter in a 400—m race accelerates uniformly for the ﬁrst 130 m and then runs with constant velocity. If the sprinter’s time for the ﬁrst 130 m is 25 5, determine (a) his acceleration, (19) his final velocity, (0) his time for the race. 11.39 In a close harness race, horse 2 passes horse 1 at point A, where the two velocities are 02 = 7 m/s and 01 = 6.8 m/s. Horse 1 later passes horse 2 at point B and goes on to win the race at point C, 400 m from A. The elapsed times from A to C for horse 1 and horse 2 are t1 = 61.5 s and t2 = 62.0 s, respectively. Assuming uniform accelerations for both horses between A and C, determine (a) the distance from A to B, (b) the position of horse 1 relative to horse 2 when horse 1 reaches the ﬁnish line C . Fig. P1139 11.40 Two rockets are launched at a fireworks performance. Rocket A is launched with an initial velocity 00 and rocket B is launched 4 5 later with the same initial velocity. The two rockets are timed to explode simulta- neously at a height of 80 m, as A is falling and B is rising. Assuming a con- stant acceleration g = 9.81 m/s2, determine (a) the initial velocity 00, (b) the velocity of B relative to A at the time of the explosion. 4.. l l i l EB_ 80m i i I I | Fig. P11.40 Problems 625 j Fig. P11.37 626 Kinematics of Particles Fig. P11.42 11.41 A police ofﬁcer in a patrol car parked in a 65 mi/h speed zone observes a passing automobile traveling at a slow, constant speed. Believing that the driver of the automobile might be intoxicated, the ofﬁcer starts his car, accelerates uniformly to 85 mi/h in 8 s, and, maintaining a constant ve- locity of 85 mi/h, overtakes the motorist 42 s after the automobile passed him. Knowing that 18 s elapsed before the ofﬁcer began pursuing the mo- torist, determine (a) the distance the ofﬁcer traveled before overtaking the motorist, (b) the motorist’s speed. 11.42 As relay runner A enters the 65-ft-long exchange zone with a speed of 42 ft/s, he begins to slow down. He hands the baton to runner B 1.82 5 later as they leave the exchange zone with the same velocity. Deter- mine (a) the uniform acceleration of each of the runners, (b) when runner B should begin to run. 11.43 In a boat race, boat A is leading boat B by 38 m and both boats are traveling at a constant speed of 168 km/h. At t = 0, the boats accelerate at constant rates. Knowing that when B passes A, t = 8 s and 0A = 228 km/h, determine (a) the acceleration of A, (b) the acceleration of B. 11.44 Car A is parked along the northbound lane of a highway, and car B is traveling in the southbound lane at a constant speed of 96 km/h. At t = 0, A starts and accelerates at a constant rate aA, while at t = 5 s, B be- gins to slow down with a constant deceleration of magnitude aA/ 6. Knowing that when the cars pass each other x = 90 m and 0A = 03, determine (a) the acceleration (1A, (1)) when the vehicles pass each other, (0) the distance be- tween the vehicles at t = 0. (ugh, = 96 km/h *- B A’UA Fig. P11.44 11.45 Two automobiles A and B traveling in the same direction in ad- jacent lanes are stopped at a trafﬁc signal. As the signal turns green, automo- bile A accelerates at a constant rate of 6.5 ft/sz. Two seconds later, automobile B starts and accelerates at a constant rate of 11.7 ft/sz. Determine (a) when and where B will overtake A, (b) the speed of each automobile at that time. 11.46 Two automobiles A and B are approaching each other in adja- cent highway lanes. At t = 0, A and B are 0.62 mi apart, their speeds are DA = 68 mi/h and 03 = 39 mi/h, and they are at points P and Q, respectively. Knowing that A passes point Q 40 s after B was there and that B passes point I P 42 s after A was there, determine (a) the uniform accelerations of A and B, {b} when the vehicles pass each other, (c) the speed of B at that time. 0A = 68 mi/h Fig. P11.46 628 Kinematics of Particles Fig. P11.55 and H156 11.51 In the position shown, collar B moves to the left with a constant velocity of 300 mm/s. Determine (a) the velocity of collar A, (b) the velocity of portion C of the cable, (0) the relative velocity of portion C of the cable with respect to collar B. Fig. P11.51 and P11.52 11.52 Collar A starts from rest and moves to the right with a constant acceleration. Knowing that after 8 s the relative velocity of collar B with re- spect to collar A is 610 mm/s, determine (a) the accelerations of A and B, (b) the velocity and the change in position of B after 6 s. 1 1.53 At the instant shown, slider block B is moving to the right with a constant acceleration, and its speed is 6 in./s. Knowing that after slider block A has moved 10 in. to the right its velocity is 2.4 in./s, determine (a) the ac- celerations of A and B, (b) the acceleration of portion D of the cable, (0) the velocity and the change in position of slider block B after 4 s. Fig. P11.53 and P11.54 11.54 Slider block B moves to the right with a constant velocity of 12 in./s. Determine (a) the velocity of slider block A, (b) the velocity of por- tion C of the cable, (a) the velocity of portion D of the cable, (d) the rela- tive velocity of portion C of the cable with respect to slider block A. 11.55 Collars A and B start from rest, and collar A moves upward with an acceleration of St2 mm/sz. Knowing that collar B moves downward with a constant acceleration and that its velocity is 200 mm/s after moving 800 mm, determine (a) the acceleration of block C, (b) the distance through which block C will have moved after 3 5. 11.56 Collar A starts from rest at t = 0 and moves downward with a constant acceleration of 180 mm/sz. Collar B moves upward with a constant acceleration, and its initial velocity is 200 Inm/s. Knowing that collar B moves through 500 mm between t = 0 and t = 2 5, determine (a) the accelerations of collar B and block C, (b) the time at which the velocity of block C is zero, (0) the distance through which block C will have moved at that time. ...
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